Calculus Examples

$\int 6 {(2 x - 1)}^{- 3} d x$ , $u = 2 x - 1$

Step 1

Let $u = 2 x - 1$ . Then $d u = 2 d x$ . Rewrite using $u$ and $d u$ .

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Step 1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{3}{u^{3}} d u$

Step 2

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$3 \int \frac{1}{u^{3}} d u$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$3 \int {(u^{3})}^{- 1} d u$

Step 3.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \int u^{3 \cdot - 1} d u$

Step 3.2.2

Multiply $3$ by $- 1$ .

$3 \int u^{- 3} d u$

Step 4

By the Power Rule, the integral of $u^{- 3}$ with respect to $u$ is $- \frac{1}{2} u^{- 2}$ .

$3 (- \frac{1}{2} u^{- 2} + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $u^{- 2}$ and $\frac{1}{2}$ .

$3 (- \frac{u^{- 2}}{2} + C)$

Step 5.1.2

Move $u^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 (- \frac{1}{2 u^{2}} + C)$

Step 5.2

Simplify.

$3 (- \frac{1}{2 u^{2}}) + C$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $- 1$ by $3$ .

$- 3 \frac{1}{2 u^{2}} + C$

Step 5.3.2

Combine $- 3$ and $\frac{1}{2 u^{2}}$ .

$\frac{- 3}{2 u^{2}} + C$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{3}{2 u^{2}} + C$

Step 6

Replace all occurrences of $u$ with $2 x - 1$ .

$- \frac{3}{2 {(2 x - 1)}^{2}} + C$

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